(Abridged) We study the properties of anisotropic and axisymmetric velocity ellipsoids from maps of the gas velocity dispersion in nearby galaxies. This data allow us to measure the azimuthal-to-radial axis ratio of gas velocity ellipsoids, which is a useful tool to study the structure of gaseous orbits in the disk. We also present the first estimates of perturbations in gas velocity dispersion maps by applying an alternative model that considers isotropic and asymmetric random motions. High-quality velocity dispersion maps of the atomic medium at various angular resolutions of the nearby spiral galaxy Messier 33, are used to test the anisotropic and isotropic velocity models. The velocity dispersions of hundreds of individual molecular clouds are also analyzed. The HI velocity dispersion of M33 is systematically larger along the minor axis, and lower along the major axis. Isotropy is only possible if asymmetric motions are considered. Fourier transforms of the HI velocity dispersions reveal a bisymmetric mode which is mostly stronger than other asymmetric motions and aligned with the minor axis of the galaxy. Within the anisotropic and axisymmetric velocity model, the stronger bisymmetry is explained by a radial component that is larger than the azimuthal component of the ellipsoid of random motions, thus by gaseous orbits that are dominantly radial. The azimuthal anisotropy parameter is not strongly dependent on the choice of the vertical dispersion. The velocity anisotropy parameter of the molecular clouds is observed highly scattered. Perturbations such as HI spiral-like arms could be at the origin of the gas velocity anisotropy in M33. Further work is necessary to assess whether anisotropic velocity ellispsoids can also be invoked to explain the asymmetric gas random motions of other galaxies.
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